Adrian, Crump, Vogt - Nonlinearity and Flight-to-Safety in the Risk-Return Tradeoff for Stocks and Bonds (wp at https://www.newyorkfed.org/research/staff_reports/sr723.html )

1. Nonlinearity and Flight-to-Safety in the Risk-Return
Tradeoff for Stocks and Bonds
Tobias Adrian Richard Crump Erik Vogt
Federal Reserve Bank of New York
The views expressed here are the authors’ and are not representative of the views of
the Federal Reserve Bank of New York or of the Federal Reserve System
June 2016
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 1

2. Introduction
Motivation: Flight-to-Safety and Nonlinearities
Investor Flight-to-Safety is pervasive in times of elevated risk
Longstaff (2004), Beber, Brandt, and Kavajecz (2009), Baele, Bekaert,
Inghelbrecht, and Wei (2013)
Theories of Flight-to-Safety predict highly nonlinear pricing functions
Vayanos (2004), Weill (2007), Caballero and Krishnamurthy (2008)
Natural question: Should Flight-to-Safety (elevated risk + nonlinear
pricing) have implications for risk-return tradeoff?
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 2

3. Introduction
Motivation: Risk-return Tradeoff
Economic theory suggests a risk-return tradeoff:
an increase in riskiness should be associated with
1. a contemporaneous drop in the asset price
2. an increase in expected returns
While 1. is easy to verify, 2. has proven hard to show
Regression of asset returns on lagged measures of risk is either
insignificant (Bekaert and Hoerova (2014), Bollerslev, Osterrieder,
Sizova, and Tauchen (2013)) or inconclusive: sometimes positive,
sometimes negative (Lundblad (2007), Brandt and Kang (2004), GJR
1993, Ghysels, Plazzi, and Valkanov (2013))
This paper: stock and bond returns reveal
Risk-return tradeoff is nonlinear
Nonlinearity is consistent with flight-to-safety
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 3

4. Introduction
Preview:
Flight-to-Safety in the Nonlinear Risk-Return Tradeoff
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 4

5. Introduction
Our Approach
1. Propose Sieve Reduced Rank Regression SRRR estimator
Rxi
t+h = ai
h + bi
h · φh(vixt) + εi
t+h, i = 1, . . . , n,
to borrow strength from the cross-section
2. Document strongly significant nonlinear risk-return tradeoff
3. Robustness to controls, subsamples, different test assets
4. Out-of-sample performance
5. Forecasting relationship within a dynamic asset pricing model
6. Theories that generate increased risk aversion as a function of
volatility provide a conceptual framework for our findings
7. Macroeconomic consequences
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 5

6. Introduction
Literature
Flight-to-safety
Vayanos (2004), Weill (2007), Caballero and Krishnamurthy (2008),
Brunnermeier and Pedersen (2009), Vayanos and Woolley (2013),
Baele, Bekaert, Inghelbrecht, and Wei (2013)
Econometric approach
Chen, Liao, and Sun (2014), Hodrick (1992), ACM(2013, 2014)
Asset return forecasting
Lettau and Van Nieuwerburgh (2008), Pesaran, Pettenuzzo, and
Timmermann (2006), Rossi and Timmermann (2010), Ang and
Bekaert (2007)
Dynamic asset pricing with stocks and bonds
Mamaysky (2002), Bekaert, Engstrom, and Grenadier (2010), Lettau
and Wachter (2010), Koijen, Lustig, and van Nieuwerburgh (2013)
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 6

7. Motivating Univariate Evidence
Outline
Motivating Univariate Evidence
Sieve Reduced Rank Regressions
Economics of Flight-to-Safety
Conclusion
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 7

8. Motivating Univariate Evidence
Motivation: Univariate Evidence
risk premium, the term spread between the 10-year and 3-month Treasury yield, and the log dividend yield.
p-values report the outcome of the joint hypothesis test under the null of no predictability. The line labeled
Linearity reports the p-values for the test of linearity in VIX, which corresponds to the null hypothesis that
the coefficients on vix2
t and vix3
t are zero. The sample consists of monthly observations from 1990:1 to
2014:9.
1-year Treasury Excess Returns
h = 6 months h = 12 months h = 18 months
V IX1 1.91 4.13 5.01 1.86 3.60 5.02 1.13 3.21 4.73
V IX2 -4.08 -4.77 -3.61 -4.86 -3.37 -4.71
V IX3 3.89 4.66 3.51 4.76 3.38 4.64
MOV E1 0.26 -0.57 -0.58 -1.94 -0.23 -2.21
MOV E2 0.17 0.81 1.00 2.14 0.54 2.43
MOV E3 -0.35 -0.97 -1.16 -2.27 -0.69 -2.57
DEF -0.99 -0.99 -1.23
VRP 2.49 2.84 3.87
TERM -1.45 -2.76 -3.47
DY 3.34 3.58 3.24
const 1.40 -3.42 -0.14 0.58 1.65 -2.75 1.05 1.57 2.17 -2.19 1.19 1.63
p-value 0.057 0.001 0.089 0.000 0.064 0.005 0.303 0.000 0.260 0.004 0.845 0.000
Linearity 0.000 0.000 0.002 0.000 0.004 0.000
Stock Excess Returns
h = 6 months h = 12 months h = 18 months
V IX1 1.00 -3.18 -2.68 0.74 -2.47 -1.98 0.78 -1.87 -1.35
V IX2 3.36 2.90 2.61 2.22 2.01 1.54
V IX3 -3.24 -2.68 -2.45 -2.15 -1.87 -1.48
MOV E1 0.15 0.27 1.14 1.26 0.13 0.33
MOV E2 -0.06 -0.23 -1.17 -1.35 -0.07 -0.41
MOV E3 -0.01 -0.01 1.20 1.23 0.16 0.40
DEF -0.58 -0.62 -0.34
VRP -2.34 -2.03 -2.33
TERM 0.47 0.94 1.03
DY 1.17 1.51 1.63
const -0.15 3.28 0.13 2.43 0.50 2.68 -0.65 2.06 0.58 2.10 0.15 2.01
p-value 0.316 0.007 0.991 0.018 0.460 0.032 0.674 0.075 0.439 0.088 0.810 0.112
Linearity 0.004 0.013 0.031 0.085 0.119 0.296
61T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 8

9. Motivating Univariate Evidence
P-values by Forecast Horizon 1990-2014
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 9

10. Motivating Univariate Evidence
P-values by Forecast Horizon 1990-2007
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 10

11. Motivating Univariate Evidence
Robustness: Nonlinearities in Realized Volatility
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 11

12. Motivating Univariate Evidence
Takeaways from the Univariate Evidence
1. Highly significant nonlinear risk-return tradeoff
2. Mirror image property for stocks and bonds
3. VIX is forecasting variable for both stocks and bonds
4. Robust to including DEF, VRP, TERM, DY
5. Robust at different horizons
6. Holds with realized downside vol in subsamples back to 1926
But: Polynomial ad-hoc
Next: What is a rigorous estimator for the nonlinearity?
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 12

13. Motivating Univariate Evidence
Generalizing the Polynomial Regression
Consider some function φi
h (vt) such that
Rxi
t+h = φi
h (vt) + εi
t+h
How can we estimate an unknown function φi
h (vt)?
Method of sieves (Chen (2007)):
Assume φi
h ∈ Φ general function space
Estimate on dim(m) approximating spaces Φm ⊂ Φm+1 ⊂ · · · ⊂ Φ
ˆφi
m,h ≡ arg min
φ∈Φm
1
T
T
t=1
(Rxi
t+h − φ (vt))2
Require m = mT → ∞ slowly as T → ∞ ⇒ ΦmT
“looks like” Φ
ˆφi
mT ,h converges to the true unknown function φi
h ∈ Φ
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 13

14. Motivating Univariate Evidence
OLS Estimation of φi
h (vt) via B-Splines
For Φm we use j = 1...m linear combinations of B-splines Bj (v)
Thus, φm,h (v) = m
j=1 γh
j · Bj (v) = γh Xm
For fixed m, estimation is via OLS: ˆγh = (XmXm)−1
XmRx
Therefore, for fixed m,
ˆφi
m,h (v) =
m
j=1
ˆγh
j · Bj (v)
We choose mT by cross-validation
B-splines are often preferred to polynomials in sieve literature:
1. Reduce collinearity concerns
2. Less oscillation at extremes
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 14

15. Motivating Univariate Evidence
Nonlinear Forecasting using SRRR Regressions 1990-2014
used in the estimation is chosen by out-of-sample cross validation. The middle plot shows a
parametric cubic polynomial regression where i
h(vt) = ai
0 + ai
1vt + ai
2v2
t + ai
3v3
t , and the right
plot shows i
h(vt) estimated by a Nadaraya-Watson kernel regression, where the bandwidth was
chosen by Silverman’s rule of thumb. The y-axis was rescaled by the unconditional standard
deviation of Rxi
t to display risk-adjusted returns.
i
h(V IX) 1990:1 to 2014:9
i
h(V IX) 1990:1 to 2007:7
50T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 15

16. Motivating Univariate Evidence
Takeaways from the Nonparametric Regressions
1. The shape of the nonlinear risk-return tradeoff is robust to
nonparametric estimation methods
2. Robust whether 2008-09 is included or not
3. Mirror image property suggests common nonlinear shape
But: Univariate approaches prone to overfitting, small samples
Next: Use cross-section to discipline inference
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 16

17. Sieve Reduced Rank Regressions
Outline
Motivating Univariate Evidence
Sieve Reduced Rank Regressions
Economics of Flight-to-Safety
Conclusion
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 17

18. Sieve Reduced Rank Regressions
SRRR: Sieve Reduced Rank Regressions
Idea: use cross-sectional information across assets to estimate φh(v)
Suppose that excess returns for i = 1, . . . , n assets are
Rxi
t+h = ai
h + bi
h · φh (vt) + f i
h zt + εi
t+h
We can rewrite this as
Rxi
t+h = ai
h + bi
h γhXm,t + f i
h zt + ˜εi
t+h
where ˜εi
t+h = εi
t+h + bi
h · (φh (vt) − γhXm,t)
Note reduced rank restriction: γh common across assets:
rank(b γh) = 1
We normalize bMarket
h = 1 as reference asset
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 18

19. Sieve Reduced Rank Regressions
Intuition for the Reduced Rank Regressions
1. Returns to each asset are regressed in the time series on lagged sieve
expansions of the VIX
2. The rank of the matrix of forecasting coefficients is reduced using an
eigenvalue decomposition, and a rank one approximation is retained
Rx
n×T
= a + b
n×1
γ
1×m
Xm
m×T
+ε
This dimensionality reduction is optimal when errors are conditionally
Gaussian and the number of regressors are fixed
The resulting factor φh(v) is a nonlinear function of volatility and is the
best common predictor for the whole cross section of asset returns
Estimation of φh(v) from a large cross-section improves identification
power relative to the univariate regressions
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 19

20. Sieve Reduced Rank Regressions
Inference
In this paper there are three primary hypotheses of interest:
H1,0 : bh = 0 H2,A : bh = 0
H2,0 : bj
h φh = 0 H1,A : bj
h φh = 0
H3,0 : φh (¯v) = 0 H3,A : φh (¯v) = 0
Proposition
vec ˆb ˆV−1
1 vec ˆb − mn
√
2mn →d,H1,0 N (0, 1)
ˆbj
2
ˆγ ˆV−1
2 ˆγ − m
√
2m →d,H2,0 N (0, 1)
ˆφh,m (¯v) − φh (¯v) ˆV3 →d,H3,0 N (0, 1) ,
Standard errors based on reverse regressions Hodrick (1992)
These are more conservative than Chen, Liao, and Sun (2014)
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 20

21. Sieve Reduced Rank Regressions
Nonlinear Forecasting using SRRR 1990-2014
Rxi
t+h = ai
+ bi
· φh(vt) + f i
h zt + εi
t+h, i = 1, . . . , n,
Table 2: Nonlinear VIX Panel Predictability: 1990 - 2014
Horizon h = 6
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.01 1.00 1.00* 1.00*** 0.31 1.00*** 0.05** 1.42*** 0.01 0.17
cmt1 0.00 0.07* 0.05* 0.07*** 0.09** 0.20*** 0.00 0.03* 0.00* 0.02***
cmt2 0.01 0.09 0.11* 0.14*** 0.15* 0.32*** 0.00 0.08** 0.00 0.02**
cmt5 0.03 0.04 0.26 0.31*** 0.25 0.60*** 0.02* 0.23** 0.01** 0.01
cmt7 0.04 0.04 0.31 0.38** 0.27 0.70*** 0.03** 0.32** 0.02** 0.00
cmt10 0.05 0.08 0.30 0.37** 0.25 0.66** 0.03** 0.39** 0.03*** 0.01
cmt20 0.08 0.22 0.39 0.49 0.23 0.74 0.05*** 0.51* 0.05*** 0.03
cmt30 0.10 0.52 0.58 0.68 0.29 0.98 0.07*** 0.70* 0.06*** 0.06
Joint p-val 0.273 0.000 0.000
Horizon h = 12
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.03 1.00 0.64 1.00* 0.09 1.00* 0.03** 0.70*** 0.00* 0.18
cmt1 0.00 0.11* 0.05 0.10*** 0.08 0.36*** 0.00 0.03** 0.00** 0.02***
cmt2 0.01 0.22 0.08 0.18** 0.12 0.57*** 0.00 0.05** 0.00 0.03***T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 21

22. Sieve Reduced Rank Regressions
Nonlinear Forecasting using SRRR 1990-2014
Rxi
t+h = ai
+ bi
· φh(vt) + f i
h zt + εi
t+h, i = 1, . . . , n,
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.01 1.00 1.00* 1.00*** 0.31 1.00*** 0.05** 1.42*** 0.01 0.17
cmt1 0.00 0.07* 0.05* 0.07*** 0.09** 0.20*** 0.00 0.03* 0.00* 0.02***
cmt2 0.01 0.09 0.11* 0.14*** 0.15* 0.32*** 0.00 0.08** 0.00 0.02**
cmt5 0.03 0.04 0.26 0.31*** 0.25 0.60*** 0.02* 0.23** 0.01** 0.01
cmt7 0.04 0.04 0.31 0.38** 0.27 0.70*** 0.03** 0.32** 0.02** 0.00
cmt10 0.05 0.08 0.30 0.37** 0.25 0.66** 0.03** 0.39** 0.03*** 0.01
cmt20 0.08 0.22 0.39 0.49 0.23 0.74 0.05*** 0.51* 0.05*** 0.03
cmt30 0.10 0.52 0.58 0.68 0.29 0.98 0.07*** 0.70* 0.06*** 0.06
Joint p-val 0.273 0.000 0.000
Horizon h = 12
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.03 1.00 0.64 1.00* 0.09 1.00* 0.03** 0.70*** 0.00* 0.18
cmt1 0.00 0.11* 0.05 0.10*** 0.08 0.36*** 0.00 0.03** 0.00** 0.02***
cmt2 0.01 0.22 0.08 0.18** 0.12 0.57*** 0.00 0.05** 0.00 0.03***
cmt5 0.02 0.33 0.18 0.39* 0.21 1.03** 0.01 0.08* 0.01 0.03
cmt7 0.02 0.35 0.23 0.48* 0.24 1.23* 0.02* 0.12* 0.01** 0.02
cmt10 0.03 0.15 0.22 0.47* 0.25 1.25* 0.02** 0.13* 0.02** 0.03
cmt20 0.05 0.12 0.31 0.65 0.27 1.52 0.04** 0.16 0.03*** 0.00
cmt30 0.06 0.17 0.44 0.88 0.33 1.92 0.05** 0.21 0.04*** 0.01
Joint p-val 0.380 0.002 0.000
Horizon h = 18
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.03 1.00 0.44 1.00 0.03 1.00 0.02** 0.59*** 0.01 0.18
cmt1 0.01 0.07 0.04 0.13*** 0.06 0.60*** 0.00 0.04*** 0.00** 0.02***
cmt2 0.01 0.19 0.07 0.24** 0.10 0.95*** 0.00 0.07*** 0.01 0.03***T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 22

23. Sieve Reduced Rank Regressions
Nonlinear Forecasting using SRRR 1990-2014
Rxi
t+h = ai
+ bi
· φh(vt) + f i
h zt + εi
t+h, i = 1, . . . , n,
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.03 1.00 0.64 1.00* 0.09 1.00* 0.03** 0.70*** 0.00* 0.18
cmt1 0.00 0.11* 0.05 0.10*** 0.08 0.36*** 0.00 0.03** 0.00** 0.02***
cmt2 0.01 0.22 0.08 0.18** 0.12 0.57*** 0.00 0.05** 0.00 0.03***
cmt5 0.02 0.33 0.18 0.39* 0.21 1.03** 0.01 0.08* 0.01 0.03
cmt7 0.02 0.35 0.23 0.48* 0.24 1.23* 0.02* 0.12* 0.01** 0.02
cmt10 0.03 0.15 0.22 0.47* 0.25 1.25* 0.02** 0.13* 0.02** 0.03
cmt20 0.05 0.12 0.31 0.65 0.27 1.52 0.04** 0.16 0.03*** 0.00
cmt30 0.06 0.17 0.44 0.88 0.33 1.92 0.05** 0.21 0.04*** 0.01
Joint p-val 0.380 0.002 0.000
Horizon h = 18
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.03 1.00 0.44 1.00 0.03 1.00 0.02** 0.59*** 0.01 0.18
cmt1 0.01 0.07 0.04 0.13*** 0.06 0.60*** 0.00 0.04*** 0.00** 0.02***
cmt2 0.01 0.19 0.07 0.24** 0.10 0.95*** 0.00 0.07*** 0.01 0.03***
cmt5 0.02 0.36 0.14 0.48* 0.18 1.65*** 0.01 0.11** 0.00 0.03**
cmt7 0.02 0.38 0.16 0.56 0.19 1.87** 0.01* 0.14** 0.00 0.03*
cmt10 0.03 0.23 0.15 0.52 0.20 1.90* 0.01* 0.15** 0.01* 0.04
cmt20 0.04 0.29 0.19 0.69 0.20 2.16 0.02* 0.15 0.02** 0.01
cmt30 0.05 0.04 0.28 0.91 0.24 2.63 0.03** 0.18 0.03** 0.00
Joint p-val 0.586 0.025 0.000
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 23

24. Sieve Reduced Rank Regressions
Nonlinear Forecasting using SRRR 1990-2007Table 3: Nonlinear VIX Panel Predictability: 1990 - 2007
Horizon h = 6
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.03 1.00 0.72 1.00 0.25 1.00 0.03 0.84* 0.00 0.12
cmt1 0.00 0.23** 0.09 0.16*** 0.15 0.59*** 0.01 0.03 0.00* 0.03***
cmt2 0.00 0.29 0.16 0.28** 0.22 0.92*** 0.01 0.10* 0.00 0.03***
cmt5 0.01 0.30 0.31 0.53* 0.34 1.57*** 0.00 0.25* 0.01 0.03*
cmt7 0.03 0.18 0.36 0.62* 0.35 1.75** 0.01 0.31* 0.02 0.02
cmt10 0.04 0.23 0.37 0.62 0.32 1.74* 0.03 0.36* 0.02* 0.02
cmt20 0.06 0.16 0.42 0.72 0.31 1.95 0.05 0.46** 0.03** 0.01
cmt30 0.05 0.27 0.51 0.85 0.35 2.25 0.06 0.58** 0.04** 0.03
Joint p-val 0.058 0.001 0.000
Horizon h = 12
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.09 1.00 0.46 1.00 0.01 1.00 0.03 0.52* 0.00 0.16
cmt1 0.00 2.09** 0.08 0.23*** 0.12 3.25*** 0.00 0.03 0.00 0.03***
cmt2 0.00 3.21* 0.13 0.40*** 0.18 5.27*** 0.00 0.07* 0.00 0.03***
cmt5 0.00 4.69 0.24 0.72** 0.31 9.32*** 0.00 0.12 0.01 0.04
cmt7 0.01 3.98 0.28 0.84** 0.33 10.77*** 0.01 0.13 0.01* 0.03
cmt10 0.03 0.71 0.27 0.80* 0.33 11.27** 0.03* 0.14 0.02** 0.04
cmt20 0.04 1.06 0.33 1.00* 0.34 13.10** 0.04** 0.14 0.03** 0.01
cmt30 0.04 1.15 0.40 1.17* 0.39 15.27** 0.06** 0.16 0.04*** 0.00
Joint p-val 0.008 0.001 0.000
Horizon h = 18
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.11 1.00 0.57 1.00** 0.13 1.00 0.03 0.54** 0.00 0.13
cmt1 0.00 0.36** 0.07 0.17*** 0.11 0.92*** 0.00 0.05** 0.00 0.03***
cmt2 0.00 0.59 0.13 0.30*** 0.19 1.55*** 0.00 0.11** 0.00 0.03***
cmt5 0.00 0.93 0.24 0.55*** 0.35 2.83*** 0.01* 0.19** 0.00 0.05*
cmt7 0.01 0.86 0.27 0.63*** 0.39 3.28*** 0.00** 0.22** 0.01* 0.05
cmt10 0.03 0.25 0.25 0.59*** 0.40 3.44*** 0.01** 0.24** 0.01** 0.06
cmt20 0.04 0.45 0.29 0.70** 0.41 3.81*** 0.02** 0.20* 0.02** 0.03
cmt30 0.03 0.43 0.37 0.85** 0.50 4.55*** 0.03** 0.25** 0.03*** 0.03
Joint p-val 0.019 0.000 0.000
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 24

25. Sieve Reduced Rank Regressions
Out-of-Sample Evidence
We form portfolios with weights ωt = V −1
t Et [Rxt+h], where risk weights V −1
t are
the unconditional variances of excess returns using data from t = 1, . . . , t
Forecast horizon and rebalancing is six months
In-sample for t = 1, . . . , t∗
, out of sample for t = t∗
, . . . , T
Horizon h = 18
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.11 1.00 0.57 1.00** 0.13 1.00 0.03 0.54** 0.00 0.13
cmt1 0.00 0.36** 0.07 0.17*** 0.11 0.92*** 0.00 0.05** 0.00 0.03***
cmt2 0.00 0.59 0.13 0.30*** 0.19 1.55*** 0.00 0.11** 0.00 0.03***
cmt5 0.00 0.93 0.24 0.55*** 0.35 2.83*** 0.01* 0.19** 0.00 0.05*
cmt7 0.01 0.86 0.27 0.63*** 0.39 3.28*** 0.00** 0.22** 0.01* 0.05
cmt10 0.03 0.25 0.25 0.59*** 0.40 3.44*** 0.01** 0.24** 0.01** 0.06
cmt20 0.04 0.45 0.29 0.70** 0.41 3.81*** 0.02** 0.20* 0.02** 0.03
cmt30 0.03 0.43 0.37 0.85** 0.50 4.55*** 0.03** 0.25** 0.03*** 0.03
Joint p-val 0.019 0.000 0.000
Table 4: Out-of-Sample Performance
Annualized Sharpe Ratio
In-Sample (1) (2) (3) (4) (5) (6) (7) (8) (9)
Cuto↵
(t⇤
/T) h(vixt) vixt Uncond. EQL MKT cmt1 cmt5 cmt10 cmt30
0.4 (Nov-1999) 1.03 0.87 0.88 0.79 0.24 1.04 0.79 0.61 0.47
0.5 (May-2002) 0.91 0.73 0.72 0.82 0.53 0.71 0.68 0.59 0.45
0.6 (Nov-2004) 1.12 0.77 0.73 0.77 0.46 0.76 0.66 0.58 0.41
3
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 25

26. Sieve Reduced Rank Regressions
Out-of-Sample Evidence
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 26

27. Sieve Reduced Rank Regressions
Summary of the SRRR Estimation
Two central empirical facts
1. Nonlinear risk-return relationship
2. Mirror image property
Shape of the nonlinear risk-return tradeoff can be robustly estimated
from the cross-section of stocks and bonds using SRRR
Robust to including common forecasting variables
Out-of-sample performance
Next: Economic Interpretation
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 27

28. Economics of Flight-to-Safety
Outline
Motivating Univariate Evidence
Sieve Reduced Rank Regressions
Economics of Flight-to-Safety
Conclusion
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 28

29. Economics of Flight-to-Safety
SRRR Estimation using Broader Test Assets
All results so far were using the market return and Treasury returns
Is the shape of the nonlinearity robust to broader asset classes?
12 industry sorted equity portfolios from Ken French
7 credit and industry sorted credit portfolios from Barclays
7 maturity sorted bond portfolios from CRSP
The broader set of test assets improves our economic insight
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 29

30. Economics of Flight-to-Safety
Expected Excess Returns across Portfolios
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 30

31. Economics of Flight-to-Safety
Industry Sorts, Treasuries, Credit: SRRR Loadings ˆbi
h
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 31

32. Economics of Flight-to-Safety
SRRR φh(vt) Separately Estimated for Stocks and Bonds
Rxi
t+h = ai
h + bi
h · φh (vt) + f i
h zt + εi
t+h, i ∈ Equities
Rxi
t+h = ai
h + bi
h · φh (vt) + f i
h zt + εi
t+h, i ∈ Treasuries
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 32

33. Economics of Flight-to-Safety
Dynamic Asset Pricing Theory
Asset pricing theory suggests that the cross-sectional intercepts ai
and slopes bi are cross-sectionally related to risk factor loadings
Et[Rxi
t+1] = αi
+ βi
(λ0 + λ1φ(vt) + Λ2xt)
where
βi
denotes a (1 × K) vector of risk factor loadings
λ0 is the (K × 1) vector of constants for the prices of risk
λ1 is the (K × 1) vector mapping φ(vt) into prices of risk
Λ2 is the (K × p) matrix of the price of risk of additional risk factors
αi
represent deviations from no-arbitrage
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 33

34. Economics of Flight-to-Safety
Dynamic Asset Pricing Estimation
We estimate
Rxi
t+1 = (αi
+ βi
λ0)
ai
+ βi
λ1
bi
φ(vt) + βi
ut+1 + εi
t+1
where
ˆφ(vt) is taken as given from the unrestricted first step regression
ut+1 represent the VAR innovations to risk factors Xt:
1. the market return
2. the one-year Treasury return
3. the nonlinear volatility factor φ(vt )
We estimate φ(vt) from a SRRR with h = 1
We use reduced rank regression approach to dynamic asset pricing
models of Adrian, Crump, and Moench (2014) to get β, λ0, λ1
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 34

35. Economics of Flight-to-Safety
Dynamic Asset Pricing: Risk Factor Exposures
first stage, (vt) is estimated from a sieve reduced rank regression Rxi
t+1 = ai
0 + bi
0 (vt)+ ei
t+1 jointly across
i. The factor innovations ut+1 = Yt+1 Et[Yt+1] are then estimated from a VAR on the market (MKT),
Treasury (TSY1), and nonlinear volatility factor ( (vt)), for vt = vixt. In a second stage, coefficients are
estimated jointly across all i = 1, . . . , n via a reduced rank regression. ***, **, and * denote statistical
significance at the 1%, 5%, and 10% level.
Exposures i
MKT
i
T SY 1
i
(v)
i
1 (↵i
+ i
0)
MKT 1.00*** 0.24*** 0.02 1.08*** 0.33***
NoDur 0.61*** 0.45 0.01 0.49** 0.21***
Durbl 1.19*** 2.24** 1.90** 0.68 0.23
Manuf 1.09*** 0.86 0.85*** 0.83*** 0.30***
Enrgy 0.71*** 1.11 1.11 0.36 0.18
Chems 0.75*** 0.80 0.20 0.87*** 0.30***
BusEq 1.44*** 1.76** 1.47*** 2.88*** 0.80***
Telcm 0.94*** 0.06 0.27 0.77* 0.25**
Utils 0.39*** 0.10 0.59 0.01 0.08
Shops 0.86*** 0.64 0.10 0.99*** 0.32***
Hlth 0.69*** 1.04 0.12 0.33 0.18**
Fin 1.08*** 1.26 0.24 0.90** 0.30***
cmt1 0.00 0.73*** 0.07 0.16** 0.03*
cmt2 0.01 1.40*** 0.14 0.32** 0.06*
cmt5 0.03* 2.90*** 0.16 0.95*** 0.19***
cmt7 0.04* 3.55*** 0.77* 1.52*** 0.32***
cmt10 0.04 3.90*** 1.19*** 1.88*** 0.41***
cmt20 0.08* 4.76*** 1.96*** 2.64*** 0.57***
cmt30 0.12** 5.45*** 2.23* 3.03*** 0.67***
AAA 0.02 2.54*** 0.68 1.11*** 0.23***
AA 0.06*** 2.28*** 0.71 1.02*** 0.21***
A 0.09*** 2.00*** 1.24*** 1.24*** 0.26***
BAA 0.12*** 1.55*** 1.58*** 1.29*** 0.26***
igind 0.08*** 1.90*** 1.67*** 1.48*** 0.31***
igutil 0.07** 1.99*** 1.73*** 1.56*** 0.33***
igfin 0.11*** 1.83*** 0.57 0.76*** 0.14**
Prices of Risk MKT TSY 1 (vt)
1 1.02*** 0.28** 0.62**
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 35

36. Economics of Flight-to-Safety
Cross-Sectional Pricing
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 36

37. Economics of Flight-to-Safety
Related Theories
Fund manager model, Vayanos (2004): manager chooses a risky
portfolio s.t. redemption risk. Generates risk aversion, liquidity
preference (flight-to-safety), and alphas.
Intermediary asset pricing, Adrian and Boyarchenko (2012):
intermediary leverage is a price of risk variable that is linked to
volatility via VaR constraints
Consumption based asset pricing, Campbell and Cochrane (2000):
surplus consumption ratio governs risk aversion. SRRR ⇒ sufficiently
high VIX corresponds with times of high risk aversion, so we ask if
VIX is related SCR?
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 37

38. Economics of Flight-to-Safety
Equilibrium Asset Pricing of Vayanos (2004)
Equilibrium expected returns are
Et Rxi
t+1 = αi
t + A (vt)Covt Rxi
t+1, RxM
t+1 + Z (vt) Covt Rxi
t+1, vt+1
where
endogenously time varying effective risk aversion A (vt)
endogenously time varying liquidity premium Z (vt)
αi
t related to transactions costs
For market,
Et RxM
t+1 = αM
t + A (vt)Vart(RxM
t+1) βM
=1
+Z (vt) Covt Rxi
t+1, vt+1
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 38

39. Economics of Flight-to-Safety
Flight-to-Safety in Global Mutual Fund Flows
Flowsi
t = ai + bi φFF (vixt) + εi
t
1.02*** 0.28** 0.62**
Table 6: Fund Flows and Nonlinear VIX
Sample: 1990 - 2014 Sample: 1990 - 2007
ai
bi
ai
bi
us equity 0.50* 1.00*** 0.29 1.00
world equity 0.88 1.77*** 1.06 3.60***
hybrid 0.94* 1.89*** 1.10 3.75***
corporate bond 0.16 0.32 0.02 0.08
HY bond 0.02 0.05 0.04 0.12
world bond 0.34 0.68 0.29 1.00
govt bond 0.63* 1.27*** 1.08 3.68***
strategic income 0.02 0.04 0.35 1.18
muni bond 0.01 0.03 0.05 0.16
govt mmmf 0.42 0.85 0.39 1.33
nongovt mmmf 0.02 0.03 0.36 1.23
national mmmf 0.00 0.00 0.16 0.56
state mmmf 0.28 0.56** 0.10 0.35
Joint p-value 0.000 0.000
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 39

40. Economics of Flight-to-Safety
Flight-to-Safety in Global Mutual Fund Flows
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 40

41. Economics of Flight-to-Safety
Intermediary VaR Constraints and Volatility
VaRi
t = ai + bi φVaR(vixt) + εi
t
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 41

42. Economics of Flight-to-Safety
Appendix: Consumption-Based Asset Pricing
Et R∗
t+1
Vart R∗
t+1
= Vart (ln Mt+1)1/2
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 42

43. Economics of Flight-to-Safety
Macroeconomic Consequences
(CUMFG), the change in goods-producing employment (LAGOODA), and the total private nonfarm payroll
series (LAPRIVA), which receive the largest weight within the Chicago Fed National Activity index. The
leading reference asset is the market excess return. (1) shows estimates of ai
h and bi
h from the panel regression
yi
t+h = ai
h + bi
hvt + "i
t+h of i’s excess returns on linear ; (2) estimates of ai
h and bi
h from the panel regression
yi
t+h = ai
h + bi
h h(vt) + "i
t+h of portfolio i’s excess return on the common nonparametric function h(·) of
vt = V IXt; (3) the same regression augmented with controls (DEFt, TERMt, DYt) · fi
described in prior
tables. The panel regressions were estimated via the sieve reduced rank regressions introduced in section 3 in
the text. ***, **, and * denote statistical significance at the 1%, 5%, and 10% level for t-statistics on ai and
fi
and for the 2
-statistic on bi
h(·) derived in Proposition 1. The joint test p-value reports the likelihood
that the sample was generated from the model where (b1
, . . . , bn
) · h(·) = 0.
Horizon h = 6
(1) Linear VIX (2) Nonlinear VIX (3) Nonlinear VIX and Controls
ai
bi
ai
bi
ai
bi
fi
DEF fi
VRP fi
TERM fi
DY
MKT 0.02 1.00 0.04 1.00 0.13 1.00 0.08*** 2.44*** 0.03* 0.40*
IPMFG 0.52*** 3.24*** 0.74 4.13*** 1.71* 5.35** 0.25* 3.34*** 0.15*** 0.34***
IP 0.52*** 3.25*** 0.74 4.10*** 1.67* 5.20*** 0.25* 3.17*** 0.15*** 0.34***
CUMFG 0.36** 2.13*** 0.51 2.74*** 1.50* 6.81*** 0.06 2.77*** 0.22*** 0.30***
LAGOODA 1.07*** 7.08*** 1.42* 8.28*** 3.56*** 9.67*** 0.49*** 2.58*** 0.19*** 0.83***
LAPRIVA 0.86*** 6.35*** 1.16 7.30*** 3.31*** 9.34*** 0.46*** 2.08*** 0.14*** 0.65***
Joint p-val 0.000 0.000 0.000
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 43

44. Conclusion
Outline
Motivating Univariate Evidence
Sieve Reduced Rank Regressions
Economics of Flight-to-Safety
Conclusion
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 44

45. Conclusion
Conclusion
1. VIX is important for joint predictability stocks and bonds
2. Forecasting relationship is nonlinear
Proposed SRRR estimator for nonlinearity and joint predictability
3. Mirror image property is evidence of flight-to-safety
Further supported by fund flows
4. Forecasting function φh(v) has a price of risk interpretation
Relates to fund manager and intermediary risk aversion
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 45

46. Appendix
Outline
Appendix
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 46

47. Appendix
Appendix: B-Spline Basis Functions
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 47

48. Appendix
Sieve, Polynomial, and Kernel Regressions
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 48

49. Appendix
Appendix: Reverse Regression Intuition
Intend to predict R
(h)
t+h = Rt+1 + Rt+2 + · · · + Rt+h,
R
(h)
t+h = ah + AhXt + εt+h, Ah = C (Rt+h, Xt) V (Xt)−1
Reverse regression sets X
(h)
t = Xt + Xt−1 + · · · + Xt−h
Rt+1 = a + AX
(h)
t + εt+1, A = C Rt+1, X
(h)
t V X
(h)
t
−1
Ah and A are related by
Ah = C R
(h)
t+h, Xt V (Xt)−1
= C Rt+1, X
(h)
t V X
(h)
t
−1
V X
(h)
t V (Xt)−1
= AV X
(h)
t V (Xt)−1
.
Under cov. station. and full rank, row i of Ah = 0 ⇔ row i of A = 0
T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 49

50. Appendix
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T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 50

51. Appendix
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T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 51

52. Appendix
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T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 52

53. Appendix
Lettau, M., and J. Wachter (2010): “The Term Structures of Equity and
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T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 53

54. Appendix
Vayanos, D. (2004): “Flight to quality, flight to liquidity, and the pricing of
risk,” Working Paper 10327, National Bureau of Economic Research.
Vayanos, D., and P. Woolley (2013): “An institutional theory of
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T Adrian, R Crump, E Vogt Nonlinear Flight-to-Safety June 2016 54